Estimating the hardness of SAT encodings for Logical Equivalence Checking of Boolean circuits

In this paper we investigate how to estimate the hardness of Boolean satisfiability (SAT) encodings for the Logical Equivalence Checking problem (LEC). Meaningful estimates of hardness are important in cases when a conventional SAT solver cannot solve a SAT instance in a reasonable time. We show that the hardness of SAT encodings for LEC instances can be estimated \textit{w.r.t.} some SAT partitioning. We also demonstrate the dependence of the accuracy of the resulting estimates on the probabilistic characteristics of a specially defined random variable associated with the considered partitioning. The paper proposes several methods for constructing partitionings, which, when used in practice, allow one to estimate the hardness of SAT encodings for LEC with good accuracy. In the experimental part we propose a class of scalable LEC tests that give extremely complex instances with a relatively small input size $n$ of the considered circuits. For example, for $n = 40$, none of the state-of-the-art SAT solvers can cope with the considered tests in a reasonable time. However, these tests can be solved in parallel using the proposed partitioning methods.